On a constant curvature statistical manifold

We will show that a statistical manifold (M, g, ∇ ) has a constant curvature if and only if it is a projectively flat conjugate symmetric manifold, that is, the affine connection ∇ is projectively flat and the curvatures satisfies R=R^* , where R^* is the curvature of the dual connection ∇ ^* . Moreover, we will show that properly convex structures on a projectively flat compact manifold induces constant curvature -1 statistical structures and vice versa.